Problem 41 - Prealgebra and Introductory Algebra

Question

Write the fractions in terms of the LCM of the denominators. $$\frac{3}{x^{2}+x-2}, \frac{x}{x+2}$$

Step-by-Step Solution

Answer

The fractions in terms of the LCM of the denominators are \(\frac{3}{(x-1)(x+2)}\) and \(\frac{x(x-1)}{(x-1)(x+2)}.\)

Step 1: Factorize the Denominator

Factorize the denominator of the first fraction \(x^{2} + x - 2\). This becomes \((x-1)(x+2)\). So, the first given fraction can be written as \(\frac{3}{(x-1)(x+2)}\). The second fraction is already in the form we need. Now, we can find the LCM of their denominators.

Step 2: Find the LCM

The LCM of denominators \(x+2\) and \((x-1)(x+2)\) is \((x-1)(x+2)\) because it is the multiple that includes all the factors present in these denominators.

Step 3: Write the Fractions in Terms of the LCM

The first fraction does not need to be changed as its denominator is the same as the LCM. The second fraction \(\frac{x}{x+2}\) can be written as \(\frac{x(x-1)}{(x-1)(x+2)}\) to express it with the LCM as the denominator. So, the answer becomes \(\frac{3}{(x-1)(x+2)}, \frac{x(x-1)}{(x-1)(x+2)}\).

Key Concepts

Fraction Addition

When it comes to adding fractions, a common denominator is crucial. It allows us to combine fractions with different denominators easily. The least common multiple (LCM) of the denominators provides this common ground. Think of the LCM as the smallest shared stage where all fractions can meet to perform their arithmetic. To carry out fraction addition effectively, you follow these steps:

• Identify the denominators of each fraction.
• Find the least common multiple (LCM) of these denominators.
• Adjust each fraction so that it has the LCM as its new denominator.
• Finally, add the adjusted fractions.

By using the LCM, you ensure the fractions are on equal footing, making it possible to add them together. This is why in the original exercise, converting fractions to a common LCM denominator was a key step before combining them.

Factoring Quadratics

In algebra, factoring quadratics is a fundamental skill, especially when working with polynomial expressions and solving equations. Quadratic expressions are usually in the form of \(ax^2 + bx + c\). Factoring involves rewriting this expression as the product of two binomial expressions. The expression can often be factored by finding two numbers that multiply to give you \(c\) (the constant term) and add to give you \(b\) (the linear coefficient). For example, in the original exercise, the expression \(x^2 + x - 2\) was factored into \((x-1)(x+2)\). These two binomials represent the roots of the quadratic equation, where the expression equals zero.Once factors are found, they can transform the expression, making it easier to compare denominators or solve equations.

Equivalent Fractions

Equivalent fractions are essential in fraction problems because they represent the same value even though their numerators and denominators might appear different. To create an equivalent fraction, you multiply both the numerator and the denominator by the same number. This rule maintains the value of the fraction, akin to resizing your piece of pie without changing how much pie you actually have.The concept is vital when manipulating algebraic fractions to have common denominators. In the exercise, we took the fraction \(\frac{x}{x+2}\) and modified it to become \(\frac{x(x-1)}{(x-1)(x+2)}\), using the LCM of the denominators as a guide. Equivalent fractions remind us that, with a little adjustment, fractions can be expressed differently while still representing the same quantity or idea. This manipulation is key to simplifying addition or subtraction across fractions.